Instability of Stabilized Finite Element Methods for the Stokes Problem in the Small Time-step Limit

报告摘要：

Recent studies indicate that consistently stabilized methods for unsteady
incompressible flows, obtained by a method of lines approach, may experience
difficulty when the time step is small relative to the spatial grid size.
Using the unsteady Stokes equations as a model problem, we show that the
semi-discrete pressure operator associated with such methods is not
uniformly coercive. We prove that for sufficiently large (relative to the
square of the spatial grid size) time steps, implicit time discretizations
contribute terms that stabilize this operator. However, we also prove that
if the time step is sufficiently small, then the fully discrete problem
necessarily leads to unstable pressure approximations. The semi-discrete
pressure operator studied in the paper also arises in pressure-projection
methods, thereby making our results potentially useful in other settings.
(Joint work with Pavel Bochev and Richard Lehoucq.)